8,027 research outputs found
Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators
Loop Quantum Gravity defines the quantum states of space geometry as spin
networks and describes their evolution in time. We reformulate spin networks in
terms of harmonic oscillators and show how the holographic degrees of freedom
of the theory are described as matrix models. This allow us to make a link with
non-commutative geometry and to look at the issue of the semi-classical limit
of LQG from a new perspective. This work is thought as part of a bigger project
of describing quantum geometry in quantum information terms.Comment: 16 pages, revtex, 3 figure
Quantum mechanics on non commutative spaces and squeezed states: a functional approach
We review here the quantum mechanics of some noncommutative theories in which
no state saturates simultaneously all the non trivial Heisenberg uncertainty
relations. We show how the difference of structure between the Poisson brackets
and the commutators in these theories generically leads to a harmonic
oscillator whose positions and momenta mean values are not strictly equal to
the ones predicted by classical mechanics.
This raises the question of the nature of quasi classical states in these
models. We propose an extension based on a variational principle. The action
considered is the sum of the absolute values of the expressions associated to
the non trivial Heisenberg uncertainty relations. We first verify that our
proposal works in the usual theory i.e we recover the known Gaussian functions.
Besides them, we find other states which can be expressed as products of
Gaussians with specific hyper geometrics.
We illustrate our construction in two models defined on a four dimensional
phase space: a model endowed with a minimal length uncertainty and the non
commutative plane. Our proposal leads to second order partial differential
equations. We find analytical solutions in specific cases. We briefly discuss
how our proposal may be applied to the fuzzy sphere and analyze its
shortcomings.Comment: 15 pages revtex. The title has been modified,the paper shortened and
misprints have been corrected. Version to appear in JHE
Choquet integrals of weighted triangular fuzzy linguistic information and their applications to multiple attribute decision making
We investigate the multiple attribute decision making problems in which attribute values take the form of triangular fuzzy linguistic information. Firstly, the definition and some operational laws of triangular fuzzy linguistic are introduced. Then, we have developed three fuzzy linguistic Choquet integral aggregation operators: fuzzy linguistic choquet ordered averaging operator, fuzzy linguistic choquet ordered geometric operator and fuzzy linguistic choquet ordered harmonic mean operator. The prominent characteristic of the operators is that they cannot only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have studied some desirable properties of these operators, such as commutativity, idempotency and monotonicity, and applied these operators to multiple attribute decision making with triangular fuzzy linguistic information. Finally an illustrative example has been given to show the developed method
Magnetic operations: a little fuzzy physics?
We examine the behaviour of charged particles in homogeneous, constant and/or
oscillating magnetic fields in the non-relativistic approximation. A special
role of the geometric center of the particle trajectory is elucidated. In
quantum case it becomes a 'fuzzy point' with non-commuting coordinates, an
element of non-commutative geometry which enters into the traditional control
problems. We show that its application extends beyond the usually considered
time independent magnetic fields of the quantum Hall effect. Some simple cases
of magnetic control by oscillating fields lead to the stability maps differing
from the traditional Strutt diagram.Comment: 28 pages, 8 figure
Constraint-wish and satisfied-dissatisfied: an overview of two approaches for dealing with bipolar querying
In recent years, there has been an increasing interest in dealing with user preferences in flexible database querying, expressing both positive and negative information in a heterogeneous way. This is what is usually referred to as bipolar database querying. Different frameworks have been introduced to deal with such bipolarity. In this chapter, an overview of two approaches is given. The first approach is based on mandatory and desired requirements. Hereby the complement of a mandatory requirement can be considered as a specification of what is not desired at all. So, mandatory requirements indirectly contribute to negative information (expressing what the user does not want to retrieve), whereas desired requirements can be seen as positive information (expressing what the user prefers to retrieve). The second approach is directly based on positive requirements (expressing what the user wants to retrieve), and negative requirements (expressing what the user does not want to retrieve). Both approaches use pairs of satisfaction degrees as the underlying framework but have different semantics, and thus also different operators for criteria evaluation, ranking, aggregation, etc
Quantum Mechanics of Extended Objects
We propose a quantum mechanics of extended objects that accounts for the
finite extent of a particle defined via its Compton wavelength. The Hilbert
space representation theory of such a quantum mechanics is presented and this
representation is used to demonstrate the quantization of spacetime. The
quantum mechanics of extended objects is then applied to two paradigm examples,
namely, the fuzzy (extended object) harmonic oscillator and the Yukawa
potential. In the second example, we theoretically predict the phenomenological
coupling constant of the meson, which mediates the short range and
repulsive nucleon force, as well as the repulsive core radius.Comment: RevTex, 24 pages, 1 eps and 5 ps figures, format change
Quantum Hall Droplets on Disc and Effective Wess-Zumino-Witten Action for Edge States
We algebraically analysis the quantum Hall effect of a system of particles
living on the disc in the presence of an uniform magnetic field
. For this, we identify the non-compact disc with the coset space
. This allows us to use the geometric quantization in order to
get the wavefunctions as the Wigner -functions satisfying a suitable
constraint. We show that the corresponding Hamiltonian coincides with the Maass
Laplacian. Restricting to the lowest Landau level, we introduce the
noncommutative geometry through the star product. Also we discuss the state
density behavior as well as the excitation potential of the quantum Hall
droplet. We show that the edge excitations are described by an effective
Wess-Zumino-Witten action for a strong magnetic field and discuss their nature.
We finally show that
LLL wavefunctions are intelligent states.Comment: 18 pages, clarifications and misprints corrected, version published
in IJGMM
Metric Properties of the Fuzzy Sphere
The fuzzy sphere, as a quantum metric space, carries a sequence of metrics
which we describe in detail. We show that the Bloch coherent states, with these
spectral distances, form a sequence of metric spaces that converge to the round
sphere in the high-spin limit.Comment: Slightly shortened version, no major changes, two new references,
version to appear on Letters in Mathematical Physic
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